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Applications of Fibonacci Numbers
Volume 3
Proceedings of 'The Third International Conference on Fibonacci Numbers and Their Applications',
Pisa, Italy, July 25-29, 1988
edited by
G. E. Bergum South Dakota State University,
Brookings, South Dakota, U.S.A.
A. N. Philippou Ministry of Education, Nicosia, Cyprus
and
A. F. Horadam University of New England,
Armidale, New South Wales, Australia
KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
Library of Congress Cataloging in Publication Data
International Conference on Fibonacci Numbers and Their Applications <3rd : 1988 Pisa, Italy)
Applications of Fibonacci numbers proceedings of the Third International Conference on Fibonacci Numbers and Their Applications, Pisa, Italy, ~uly 25-29, 1988 / edited by G.E. Bergum, A.N. Phil ippou, and A.F. Horadam.
p. cm. "Volume 3." Includes bibliographical references. ISBN 0-7923-0523-X 1. Fibonacci numbers--Congresses. I. Bergum, Gerald E.
II. Phi 1 ippou, Andreas N. III. Horadam, A. F. IV. Title. OA241.I58 1988 512' .72--dc20 89-24547
ISBN-13: 978-94-010-7352-3 e-ISBN-13: 978-94-009-1910-5 001: 10.1007/978-94-009-1910-5
Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
Kluwer Academic Publishers incorporates the publishing programmes of
D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press.
Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers,
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Cover figure by Ms Sabine Lohmann
printed on acid free paper
All Rights Reserved © 1990 by Kluwer Academic Publishers
Softcover reprint of the hardcover 1 st edition 1990 No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
TABLE OF CONTENTS
A REPORT ON THE THIRD INTERNATIONAL CONFERENCE ... LIST OF CONTRIBUTORS TO THIS PROCEEDINGS FOREWORD THE ORGANIZING COMMITTEES LIST OF CONTRIBUTORS TO THE CONFERENCE INTRODUCTION
VII
ix xv
XVll
XIX
XXlll
TIlE ROLE OF THE FIBONACCI SEQUENCE IN THE ISOLATION OF THE REAL ROOTS OF POLYNOMIAL EQUATIONS
A. G. Akritas & P. G. Bradford .......................................... 1 A GCD PROPERTY ON PASCAL'S PYRAMID AND TIlE CORRESPONDING LCM PROPERTY OF THE MODIFIED PASCAL PYRAMID
Shiro Ando & Daihachiro Sato ........................................... 7 TRANSLATABLE AND ROTATABLE CONFIGURATIONS WHICH GIVE EQUAL PRODUCT, EQUAL GCD AND EQUAL LCM PROPERTIES SIMULTANEOUSLY
Shiro Ando & Daihachiro Sato .......................................... 1.5 FIBONACCI LENGTH OF GENERATING PAIRS IN GROUPS
C. M. Campbell, H. Doostie & E. F. Robertson ............................... 27 A GENERALIZATION OF FIBONACCI TI~EES
Renato M. Capocelli ................................................ 37 GENERALIZED FIBONACCI NUMflEHS ARE ROUNDED POWERS
Renato M. Capocelli & Paul Cull ........................................ 57 ON GENERALIZED FIBONACCI NUMBERS OF GRAPHS
Michael Drmota .................................................... 63 AN INVESTIGATION OF SEQUENCES DERIVED FROM HOGGATT SUMS AND HOGGATT TRIANGLES
Daniel C. Fielder & Cecil O. Alford ....................................... 77 REPRESENTATION OF NATURAL NUl\IflERS AS SUMS OF FIBONACCI NUMBERS: AN APPLICATION TO MODERN CRYPTOGRAPHY
Piero Filipponi & Emilio Montolivo ........................................ 89 A KOTE ON RAMIFICATIONS CONCERNIKG TIlE CONSTRUCTION OF PYTHAGOREAN TRIPLES FROM RECURSIVE SEQUENCES
Herta T. Freitag ................................................... 101
ON TIl E REPRESENTATION OF {F kn/F'n }, {Fkn /L11}' {Lkn/Ln}, AND {Lkn/Fn } AS ZECKENDORF SU.\IS
Herta T. Freitag ................................................... 107 FUNCTIONAL RECURRENCES
Krystyna Grytczuk & Aleksander Grytczuk ................................ 115 CONCENTRIC CYCLES IN MOSAIC GRAPIlS
Heiko Harborth .................................................... 123 FlI30NACCI TRIANGLES
Heiko Harborth & Arnfried Kemnitz ...................................... 129 l\WSAIC NUMBERS OF FIBONACCI TREES
Heiko Harborth & Sabine Lohmann . ...................................... 133 FALLING FACTORIAL POLYNOMIALS OF' GENERALIZED FIBONACCI TYPE
A. F. Horadam .................................................... 139
VI TABLE OF CONTENTS
SOME NOTES ON FIBONACCI BINARY SEQUENCES Yasuichi Horibe .................................................... 155
CONGRUENCES FOR WEIGHTED AND DEGENERATE STIRLING Nm,1BERS F. T. Howard ..................................................... 161
AN INVERSE THEOREM ON FIBONACCI NUMBERS Naotaka Imada ............................. • ........ • ........ • .... 171
SOME RESULTS ON DIVISIBILITY SEQUENCES Norbert Jensen . .................................................... 181
ON l\!EKTAL CALCULATION OF REPEATING DECIMALS, FINDING FIBONACCI NUMBERS AND A CONNECTION TO PASCAL'S TRIANGLE
Marjorie Bicknell-Johnson . ............................................ 191 DIOPHAKTINE REPRESENTATION OF FIBONACCI NUMBERS OVER NATURAL NUMBERS
James P. Jones . ................................................... 197 ON PRIME DIVISORS OF THE TERMS OF SECOND ORDER LINEA R RECURRENCE SEQUEKCES
Peter Kiss ....................................................... 203 AN ALTERNATING PRODUCT REPRESENTATION FOR REAL KUMBERS
Arnold Knopfmacher & John Knopfmacher .................................. 209 MAXIMUM LENGTH OF THE EUCLIDEAN ALGORITHM AND CONTINUED FRACTIONS IN IF(X)
Arnold Knopfmacher & John Knopfmacher .................................. 217 RECURRENCE RELATIONS IN SINUSOIDS AND THEIR APPLICATIONS TO SPECTRAL ANALYSIS AND TO THE RESOLUTION OF ALGEBRAIC EQUATIONS
Joseph Lahr ...................................................... 223 A RECURRENCE RELATION FOR GAUSSIAN MULTINOMIAL COEFFICIENTS
S. L. Lee & G. M. Phillips . ............................................ 239 SOME BINOMIAL FIBONACCI IDENTITIES
Calvin T. Long .................................................... 241 A SURVEY OF PROPERTIES OF THIRD ORDER PELL DIAGONAL FUNCTIONS
Br. 1. M. Mahon & A. F. Horadam ....................................... 2.5.5 MULTIVARIATE FIBONACCI POLYNOMIALS OF ORDER K AND THE MULTIPARAMETER NEGATIVE BINOMIAL DISTRIBUTION OF THE SAME ORDER
Andreas N. Philippou & Demetris L. Antzoulakos .............................. 273 LONGEST CIRCULAR RUNS WITH AN APPLICATION IN RELIABILITY VIA THE FIBONACCI-TYPE POLYNOMIALS OF ORDER K
Andreas N. Philippou & Frosso S. Makri . ................................... 281 FIBONACCI NUMBERS AND AN ALGORITH;VI OF LEMOINE AND KATAI
Jukka Pihko ...................................................... 287 GENERALIZATIOKS OF SEQUENCES OF LUCAS AND BELL
A. G. Shannon & A. F. Horadam . ........................................ 299 DISTRIBUTION OF RESIDUES OF CERTAIN SECOND-ORDER LINEAR RECURRENCES MODULO P
Lawrence Somer ..............................................•.... 311 THE FIBONACCI TREE, HOFSTADTER AND THE GOLDEN STRIKG
Keith Tognetti, Graham Winley & Tony van Ravenstein ......................... 325 THREE NUMBER TREES - THEIR GROWTH RULES AND RELATED NUMBER PROPERTIES
1. C. Turner ..................................................... 335 SUBJECT INDEX ...................................... . .............. 351
A REPORT ON THE THIRD INTERNATIONAL CONFERENCE
ON FIBONACCI NUMBERS AND THEIR APPLICATIONS
A newspaper article at Pisa, Italy, with a prominent headline: "CONVEGNO PARLANO I MATEMATICI L'INCONTRO IN OMMAGIO A FIBONACCI" heralded our Third International Conference on Fibonacci Numbers and Their Applications which was held in Pisa, Italy, July 25th-29th, 1988. A stamp: "I NUMERI DI FIBONACCI CONGRESSO INTERNAZIONALE, 26-7-1988" commemorated it.
Of course, mathematicians all across the globe, and especially those who are so fortunate as to have become interested in "Fibonacci-type mathematics," had known about it for some time. The August 1987 issue of The Fibonacci Quarterly had brought the glad tidings: an announcement that our third conference was to take place at the University of Pisa during the last week of July 1988.
By mid June 1988, we held the coveted program in our hands. 66 participants were listed, and they came from 22 different countries, the U.S. heading the list with a representation of 20, followed by Italy and Australia. Of course, it was to be expected that at conference time proper additional names would lengthen the count. Forty-five papers were to be presented, several of them with coauthors; there were 3 women speakers.
Theoretically sounding titles abounded. There was Andreas N. Philippou's paper, coauthored by Demetris L. Antzoulakes: "Multivariate Fibonacci Polynomials of Order K and the Multiparameter Negative Binomial Distribution of the Same Order." But, rather intriguingly, practical interests wedged themselves in also with Piero Filipponi's paper, coauthored by Emilio Montolivo: "Represent.ation of Natural Numbers as a Sum of Fibonacci Numbers: An Application to Modern Cryptography." This again highlighted one of the joys mathematicians experience: the interplay between theoretical and applied mathematics.
What a delight it was to meet in Pisa, Italy, the birthplace of Leonardo of Pisa, son of Bonacci, "our" Fibonacci (=1170-1250). We already knew that-befittingly, and much to our pleasure- Pisa had honored its mathematical son by a st.atue. My friends and I were among the many (maybe it was all of them) who made a pilgrimage to Fibonacci's statue. It was a fairly long walk, eventually on Via Fibonacci(!), along the Arno River, until we finally found him in a pretty little park. He seemed thoughtful, and appeared to enjoy the sight of the nearby shrubs and flowers. I felt like thanking him for "having started it all," for having coined the sequence that now bears his name. It would have been nice to invite him to our sessions. I predict he would have been thoroughly startled. What had happened since 1202 when his Liber Abaci was published?!
Almost invariably, the papers were of very high caliber. The great variety of topics and the multitude of approaches to deal with a given mathematical idea was remarkable and rather appealing. And it was inspiring to coexperience the deep involvement which authors feel with their topic.
Vll
viii A REPORT ON ...
We worked hard. The sessions started at 9 a.m. and with short intermissions (coffee break and lunch) they lasted till about 5:30 p.m. As none of the papers were scheduled simultaneously, we could experience the luxury of hearing ALL presentations.
We did take out time to play. Of course, just to BE in Pisa was a treat. We stepped into the past, enwrapped into the charm of quaint, old buildings, which -could they only talk-would fascinate us with their memories of olden times. As good fortune would have it (or, was it the artistry of Roborto Dvornicich, Professor of Mathematics at the University of Pisa, who arranged housing for the conference participants) my friends and I stayed at the Villa Kinzica-across the street form the Leaning Tower of Pisa. Over a plate of spaghetti, we could see that tower, one of the "seven wonders of the world~ whose very construction took 99 years. And-it REALLY leans! We were charmed by the seven bells, all chiming in different tones. But-most of all-we pictured Galileo Galilei excitedly experimenting with falling bodies ...
I would be amiss if I did not mention the Botanical Garden of Pisa-situated adjacent to our conference room at POLO DIDATTICO DELLA FACOLTA DI SCIENZE. In the summer of 1543 (the University of Pisa itself was founded in the 12th century) this garden was opened as the first botanical garden in Western Europe. Its present location was taken up 50 years later. While we may not have been able to recognize "METASEQUOIA GLYPTOSTROBOIDES" the peace and serenity of this beautiful park struck chords in all of us.
On the third day, the Conference terminated at noon, and we took the bus to Volterra. The bus ride itself ushered in a trip long to be remembered. The incredibly luscious fields of sunflowers and sunflowers-an actual ocean of yellows-were not only joyous, but also touched our mathematical souls. Do Fibonacci numbers not play an important role in deciphering nature's handiwork in sunflowers?
Volterra, situated about 550 metres above sea-level, immediately transplanted us into enigmatic Etruscan, as well as into problematic Medieval times. While we were fascinated both by the historic memorabilia, as well as by the artifacts and master pieces, the magnificent panorama of the surrounding landscape enhanced our enjoyment still further.
As has become tradition in our conference, a banquet was held on the last night before the closing of our sessions. Lucca, the site of the meeting, provided a wonderful setting for a memorable evening, Ligurian in origin, it bespeaks of Etruscan culture, and exudes the charm of an ancient city.
The spirit at the banquet highlighted what had already become apparent during the week: that the Conference had not only been mind-streatching, but also heartwarming. Friendships which had been started, became knitted more closely. New friendships were formed. The magnetism of common interest and shared enthusiasm wove strong bonds among us. \'Ve had corne from different cultural and ethnic backgrounds, and our native tongues differed. Yet, we truly understood each other. And we cared for each other.
I believe, I speak for all of us if I express by heartfelt thanks to all members of the International, as well as of the Local Committee whose dedication and industriousness gave us this unforgettable event. Our gratitude also goes to the University of Pisa whose generous hospitality we truly appreciated. I would also like to thank all participants, without whose work we could not have had this treat.
"Auf Wiedersehen~ then, at Conference number Four in 1990.
Herta T. Freitag
LIST OF CONTRIBUTORS TO THIS PROCEEDINGS
PROFESSOR A. G. AKRITAS (pp. 1-6) Computer Science Department The University of Kansas 110 Strong Hall Lawrence, KS 66045-2192
PROFESSOR CECIL O. ALFORD (pp. 77-88) School of Electrical Engineering Georgia Institute of Technology Atlanta, Georgia 30332-0250
PROFESSOR SHIRO AN DO (pp. 7-14; 15-26) College of Engineering Hosei University 3-7-2, Kajino-Cho Koganei-shi Tokyo 184, Japan
MR. DEMETRIS L. ANTZOULAKOS (pp.273-279) Department of Mathematics University of Patras 261.10 Patras, Greece
MR. P. G. BRADFORD (pp. 1-6) Computer Science Department The University of Kansas 110 Strong Hall Lawrence, KS 66045-2192
DR. C. M. CAl\IPBELL (pp. 27-35) The Mathematical Institute University of St. Andrews The North Haugh St. Andrews KY16 9SS Fife, Scotland
PROFESSOR RENATO M. CAPOCELLI (pp. 37-56; 57-62) Dipartimento di Matematica Universita di Roma "La Sapienze~ 00815 Roma, Italy
IX
x
PROFESSOR PAUL CULL (pp. 57-62) Department of Computer Science Oregon State University Corvallis, Oregon 97331
MR. H. DOOSTIE (pp. 27-35) Department of Mathematics University for Teacher Education 49 Mofateh A venue Tehran 15614 Iran
DR. MICHAEL DRMOTA (pp.63-76) Department of Discrete Mathematics Technical University of Vienna Wiedner Hauptstrasse 8-10/118 A-1040 Vienna, Austria
PROFESSOR DANIEL C. FIELDER (pp. 77-88) School of Electrical Engineering Georgia Institute of Technology Atlanta, Georgia 30332-0250
MR. PIERO FILIPPONI (pp. 89-99) Fondazione Ugo Bordoni Viale Baldassarre Castiglione, 59 00142-Roma, Italy
CONTRIBUTORS TO THIS PROCEEDINGS
PROFESSOR HERTA T. FREITAG (PP. 101-106; 107-114) B-40 Friendship Manor 320 Hershberger Road, N.W. Roanoke, Virginia 24012
PROFESSOR ALEKSANDER GRYTCZUK (pp. 115-121) 65-562 Zielona Gora UL. Sucharskiegs 18/14 Poland
PROFESSOR KRYSTYNA GRYTCZUK (pp. 115-121) 65-562 Zielona Gora UL. Sucharskiegs 18/14 Poland
DR. IIEIKO HARBORTH (pp. 123-128; 129-132; 133-138) Bienroder Weg 47 D-3300 Braunschweig \Vest Germany
CONTRIBUTORS TO THIS PROCEEDINGS
PROFESSOR A. F. HORADAM (pp. 139-153; 255-271; 299-309) Department of Math., Stat., & Compo Sci. University of New England Armidale, N.S.W. 2351 Australia
PROFESSOR YASUICHI HORIBE (pp. 155-160) Department of Information Systems Faculty of Engineering Shizuoka University Hamamatsu 432, Japan
PROFESSOR F. T. HOWARD (pp.161-170) Department of Mathematics and Computer Science Box 7311, Reynolda Station Wake Forest University Winston-Salem, NC 27109
PROFESSOR NAOTAKA IMADA (pp.171-179) Department of Mathematics Kanazawa Medical University Uchinada, Ishikawa 920-02 Japan
MR. NORBERT JENSEN (pp. 181-189) Mathematisches Seminar Der Christian-Albrccht.s-Univ. Zu Kiel Ludewig Meyn-St.r. 4 D-2300 Kiel 1, F.R. Germany
DR. MARJORIE BICKNELL-JOHNSON (pp. 191-195) 665 Fairlane Avenue Santa Clara, CA 95051
PROFESSOR JAMES P. JONES (pp. 197-201) Department of Math. and Stat. University of Calgary Calgary (T2N 1N4) Alberta, Canada
DR. ARNFRIED KEMNITZ (pp. 129-132) Wiimmeweg 10 3300 Braunschweig West Germany
DR. PETER KISS (pp. 203-207) 3300 Eger Csiky S. U. 7 mfsz. 8 Hungary
xi
xii CONTRIBUTORS TO THIS PROCEEDINGS
DR. ARNOLD KNOPFMACIlER (pp. 209-216; 217-222) Department of Computational and Applied Mathematics University of the \Vitwatersrand 1 J an Smuts A venue Johannesburg, South Africa 2050
PROFESSOR JOHN KNOPFMACHER (pp. 209-216; 217-222) Department of Mathematics University of the Witwatersrand Johannesburg, South Africa 2050
DR. JOSEPH LAHR (pp. 223-238) 14, Rue Des Sept Arpents L-1139 Luxembourg Grand Duchy of Luxembourg Luxembourg
PROFESSOR S. L. LEE (pp. 239-240) Department of Mathematics National University of Singapore Singapore 0511, Singapore
MRS. SABINE LOHMANN (pp.133-138) H. Billtenweg 7 0-3300 Braunschweig West Germany
PROFESSOR CALVIN T. LONG (pp.241-254) Department of Mathematics Washington State University Pullman, WA 99164-2930
BR. J. M. MAHON (pp. 255-271) 12 Shaw Avenue Kingsford N.S.W. 2032 Australia
MR. FROSSO S. MAKRI (pp. 281-286) Department of Mathematics University of Patras Patras, Greece
DR. EMILIO MONTOLIVO (pp. 89-99) Fondazione Ugo Bordoni Viale Baldassarre Castiglione, 59 00142-Roma, Italy
CONTRIBUTORS TO TIllS PROCEEDINGS
PROFESSOR ANDREAS N. PHILIPPOU (pp. 273-279; 281-286) Minister of Education Ministry of Education Nicosia, Cyprus
PROFESSOR G. M. PHILLIPS (pp. 239-240) The Mathematical Institute University of St. Andrews The North Haugh St. Andrews KY16 9SS Fife, Scotland
DR. JUKKA PIHKO (pp. :287-297) University of Helsinki Department of l\lathematics HaIIituskatu 15 SF-OOIOO Helsinki, Finland
DR. E. F. ROBERTSON (pp. 27-35) University of St. Andrews The Mathematical Institute The North Haugh St. Andrews KY16 9SS Fife, Scotland
PROFESSOR DAIHACHIRO SATO (pp. 7-14; 15-26) Department of Mathematics and Statistics University of Regina Regina, Saskatchewan Canada, S4S OA2
PROFESSOR A. G. SHANNON (pp.299-309) University of Technology, Sydney School of Mathematical Sciences P.O. Box 123 Broadway N.S.W. 2007 Australia
DR. LA WRENCE SOMER (pp. 311-324) 1400 20th St., NW #619 Washington, D.C. 20036
DR. KEITH TOGNETTI (pp. 325-334) Department of Mathematics University of WoIIongong P.O. Box 1144 WoIIongong, 2500 Australia
Xlii
xiv
PROFESSOR J. C. TURNER (pp.335-350) School of Math. & Compo Sci. University of Waikato Private Bag Hamilton, New Zealand
DR. TONY V AN RA VENSTEIN (pp. 325-334) Department of Mathematics University of Wollongong P.O. Box 1144 Wollongong, 2500 Australia
DR. GRAHAM WINLEY (pp.325-334) Institute for Advanced Education University of Wollongong P.O. Box 1144 Wollongong, 2500 Australia
CONTRIBUTORS TO THIS PROCEEDINGS
FOREWORD
This book contains thirty-six papers from among the forty-five papers presented at the Third International Conference on Fibonacci Numbers and Their Applications which was held in Pisa, Italy from July 25 to July 29, 1988 in honor of Leonardo de Pisa. These papers have been selected after a careful review by well known referees in the field, and they range from elementary number theory to probability and statistics. The Fibonacci numbers are their unifying bond.
It is anticipated that this book, like its two predecessors, will be useful to research workers and graduate students interested in the Fibonacci numbers and their applications.
August 1989
xv
The Editors
Gerald E. Bergum South Dakota State University Brookings, South Dakota, U.S.A.
Andreas N. Philippou Ministry of Education Nicosia, Cyprus
Alwyn F. Horadam University of New England Armidale N.S.W., Australia
THE ORGANIZING COMMITTEES
LOCAL COMMITTEE
Dvornicich, Roberto, Chairman
Filipponi, Piero
Perelli, Alberto
Viola, Carlo
Zannier, Umberto
INTERNATIONAL COMMITTEE
Horadam, A. F. (Australia), Co-chairman
Philippou, A. N. (Cyprus), Co-chairman
Ando, S. (Japan)
Bergum, G. E. (U.S.A.)
Johnson, M. B. (U.S.A.)
Kiss, P. (Hungary)
Tijdeman, Robert (The Netherlands)
Tognetti, K. (Australia)
XVII
LIST OF CONTRIBUTORS TO THE CONFERENCE'
ADLER, I., RR 1, Box 532, North Bennington, VT 05257-9748. "Separating the Biological from the Mathematical Aspects of Phyllotaxis."
*AKRITAS, A. G., (coauthor P. G. Bradford). "The Role of the Fibonacci Sequence in the Isolation of the Real Roots of Polynomial Equations."
*ALFORD, C. 0., (coauthor D. C. Fielder). "An Investigation of Sequences Derived From Hoggatt Sums and Hoggatt Triangles."
*ANDO, S., (coauthor D. Sato). "A GCD Property on Pascal's Pyramid and the Corresponding LCM Property of the Modified Pascal Pyramid."
*ANDO, S. (coauthor D. Sato). "Translatable and Rotatable Configurations Which Give Equal Product, Equal GCD and Equal LCM Properties Simultaneously."
*ANTZOULAKOS, D. L., (coauthor A. N. Philippou). "Multivariate Fibonacci Polynomials of Order k and the Multiparameter Negative Binomial Distribution of the Same Order."
BENZAGHOU, D., Universite Des Sciences Et De La, Technologie Houari Boumediene, Institut de Mathematiques, El-Alia, B. P. No. 32, Bab Ezzouar, Alger. "Linear Recurrences with Polynomial Coefficients."
*BRADFORD, P. G., (coauthor A. G. Akritas). "The Role of the Fibonacci Sequence in the Isolation of the Real Roots of Polynomial Equations."
*CAMPBELL, C. M., (coauthors H. Doostie and E. F. Robertson). "Fibonacci Length of Generating Pairs in Groups."
CAMPBELL, C. M., (coauthors E. F. Robertson and R. M. Thomas). "A Fibonacci-Like Sequence and its Application to Certain Problems in Group Presentations."
*CAPOCELLI, R. M. "A Generalization of Fibonacci Trees." *CAPOCELLI, R. M., (coauthor P. Cull). "Generalized Fibonacci Numbers are Rounded
Powers." *CULL, P., (coauthor R. Capocelli). "Generalized Fibonacci Numbers are Rounded
Powers." *DOOSTIE, H., (coauthors C. M. Campbell and E. F. Robertson). "Fibonacci Length of
Generating Pairs in Groups." *DRMOTA, M. "On Generalized Fibonacci Numbers of Graphs." *FIELDER, D. C., (coauthor C. O. Alford). "An Investigation of Sequences Derived From
Hoggatt Sums and Hoggatt Triangles." *FILIPPONI, P., (coauthor E. Montolivo). "Representation of Natural Numbers as Sums of
Fibonacci Numbers: An Application to Modern Cryptography." *FREITAG, H. T. "A Note on Ramifications Concerning the Construction of Pythagorean
Triples From Recursive Sequences." *FREITAG, II. T. "On the Representation of {F kn/F n }, {F kn/Ln}, {Lkn/Ln}, and {Lkn/F n}
as Zeckendorf Sums."
*The asterisk indicates that the paper is included in this book and that the author's address can be found in the List of Contributors to the Proceedings. The address of an author follows the name if the article does not appear in this book.
XIX
x.x CONTIUBUTORS TO THE CONFERENCE
*GRYTCZUK, A. "Functional Recurrences." *GRYTCZUK, K. "Functional Recurrences." *IIARBORTH, H. "Concentric Cycles in Mosaic Graphs." *IIARBORTH, H., (coauthor A. Kemnitz). "Fibonacci Triangles." *IIARBORTH, B., (coauthor S. Lohmann). "l\Iosaic Numbers of Fibonacci Trees."
HINDIN, H. J., Engineering Technologies Group, Suite 202,5 Kinsella StrecL, Dix Hills, NY, 11746. "Inverse Figurate Numbers, Difference Triangles, and the Beta Function."
*BORADAM, A. F. "Falling Factorial Polynomials of Generalized Fibonacci Type." *HORADAM, A. F., (coauthor Br. J. M. Mahon). "A Survey of Properties of Third Order Pell
Diagonal Functions." *HORADAM, A. F., (coauthor A. G. Shannon). "Generalizations of Sequences of Lucas and
Bell." IIORADAM, A. F. "Light in the Darkness: Fibonacci of Pisa."
*HORIBE, Y. "Some Notes on Fibonacci Binary Sequences." *IIO\VARD, F. T. "Congruences for \Veighted and Degenerate Stirling Numbers." *IMADA, N. "An Inverse Theorem on Fibonacci Numbers." *.JENSEN, N. "Some Results on Divisibility Sequences." *JOIINSON, M. "On Mental Calculation of Repeating Decimals, Finding Fibonacci Numbers and
a Connection to Pascal's Triangle." *.JONES, J. P. "Diophantine Representation of Fibonacci Numbers Over Natural NUlllbers." *KEMI\ITZ, A., (coauthor H. Barborth). "Fibonacci Triangles." *KISS, P. "On Prime Divisors of the Terms of Second Order Linear Recurrence Sequences." *KI\OPFMACHER, A., (coauthor J. Knopfmacher). "'An Alternating Product. Representation
for Real Numbers." *KNOPFMACHER, A., (coauthor J. Knopfmacher). "Maximulll Length of the Euclidean
Algorithm and Continued Fractions in IF(X)." *KNOPFMACHER, J., (coauthor A. Knopfmacher). "An Alternating Product Representation
for Real Numbers." *KI\OPFMACHER, J., (coauthor A. Knopfmacher). "Maximum Length of the Euclidean
Algorithm and Continued Fractions in IF(X)." *LAIIR, J. "Recurrence Relations in Sinusoids and Their Applications to Spectral Analysis and
to the Resolution of Algebraic Equations." *LEE, S. L., (coauthor G. M. Phillips). "A Recurrence Relat.ion for Gaussian :\Iultinomial
Coefficien ts." LEEB, K., Universit.y of Erlangen-Nurnberg, Inst. fur Yfath., Martensstrabe 3, (8520) Erlangen,
Federal Republic of Germany. "A Class of Piecewise Linear Transformations with Moderate Growth but Absolutely Absurd Behavior Under Iteration."
*LOBMANN, S., (coauthor II. IIarborth). "Mosaic Numbers of Fibonacci Trees." *LONG, C. T. "Some Binomial Fibonacci Identit.ies." *l\L\IION, BR. J. i\L, (coauthor A. F. IIoradam). "A Survey of Properties of Third Order Pell
Diagonal Functions." *MAKRI, F. S., (coauthor A. N. Philippou). "Longest Circular Runs with an Application in
Reliability via the Fibonacci-Type Polynomials of order k." *MONTOLIVO, K, (coauthor P. Filipponi). "Representation of Nat.ural Numbers as Sums of
Fibonacci Numbers: An Application to Modern Cryptography." *PHILIPPOU, A. N., (coauthor D. L. Antzoulakos). "Multivariate Fibonacci Polynomials of
Order k and the MuJt.iparamcter Negat.ive Binomial Distribution of t.he Same Order." *PIIILIPPOU, A. N., (coauthor F. S. Makri). "Longest. Circular RUIIS with an Application in
Reliability via the Fibonacci-Type Polynomials of order k."
CONTRIBUTORS TO THE CONFERENCE
*PHILLIPS, G. M., (coauthor S. 1. Lee). "A Recurrence Relation for Gaussian Multinomial Coefficients."
*PIHKO, J. "Fibonacci Numbers and an Algorithm of Lemoine and Katai ." POPOV, B. S., Macedonian Academy of Sciences and Arts, Research Center for New
Technology, AV. "Krste Misirkov" BB, POB 428, 91000 Skopje, Yugoslavia. "Some Accelerations of the Convergence of a Certain Class of Sequences."
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ROBBINS, N., Department of Mathematics, San Francisco State University, San Francisco, CA 94132. "Lucas Numbers of the Form px2, where p is Prime."
*ROBERTSON, E. F., (coauthors C. M. Campbell and H. Doostie). "Fibonacci Length of Generating Pairs in Groups."
ROBERTSON, E. F., (coauthors C. M. Campbell and R. M. Thomas). "A Fibonacci-Like Sequence and its Application to Certain Problems in Group Presentations."
*SATO, D., (coauthor S. Ando). "A GCD Property on Pascal's Pyramid and the Corresponding LCM Property of the Modified Pascal Pyramid."
*SATO, D., (coauthor S. Ando). "Translatable and Rotatable Configurations Which Give Equal Product, Equal GCD and Equal LCM Properties Simultaneously."
*SIIANNON, A. G., (coauthor A. F. Horadam) . "Generalizations of Sequences of Lucas and Bell."
*SOMER, 1. "Distribution of Residues of Certain Second-Order Linear Recurrences Modulo p." THOMAS, R. M., Department of Comput.ing Studies, University of Leicester, Leicester LEI
7RH, England, (coauthors C. M. Campbell and E. F. Robertson). "A Fibonacci-Like Sequence and its Application to Certain Problems in Group Presentations."
*TOGNETTI, K., (coauthors G. Winley and T. van Ravenstein). "The Fibonacci Tree, Hofstadter and the Golden String."
*TURNER, J. C. "Three Number Trees-Their Growth Rules and Related Number Properties." *VAN RAVENSTEIN, T. , (coauthors K. Tognetti and G. Winley). "The Fibonacci Tree,
Hofstadter and the Golden String." *WINLEY, G., (coauthors K. Tognetti and T. van Ravenstein). "The Fibonacci Tree,
Hofstadter and t.he Golden String."
INTRODUCTION
The numbers
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... ,
known as the Fibonacci numbers, have been named by the nineteenth-century French mathematician Edouard Lucas after Leonard Fibonacci of Pisa, one of the best mathematicians of the Middle Ages, who referred to them in his book Liber Abaci (1202) in connection with his rabbit problem.
The astronomer Johann Kepler rediscovered the Fibonacci numbers, independently, and since then several renowned mathematicians have dealt with them. We only mention a few: J. Binet, B. Lame, and E. Catalan. Edouard Lucas studied Fibonacci numbers extensively, and the simple generalization
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ..
bears his name.
During the twentieth century, interest in Fibonacci numbers and their applications rose rapidly. In 1961 the Soviet mathematician N. Vorobyov published Fibonacci Numbers, and Verner E. Hoggatt, Jr., followed in 1969 with his Fibonacci and Lucas Numbers. Meanwhile, in 1963, Hoggatt and his associates founded The Fibonacci Association and began publishing The Fibonacci Quarterly. They also organized a Fibona.cci Conference in California, U.S.A., each year for almost sixteen years until 1979. In 1984, the First International Conference on Fibonacci Numbers and Their Applications was held in Patras, Greece, and the proceedings from this conference have been published. It was anticipated at that time that this conference would set the beginning of international conferences on the subject to be held every two or three years in different countries. With this intention as a motivating force, The Second International Conference on Fibonacci Numbers and Their Applications was held in San Jose, California, U.S.A., August 13-16, 1986. The proceedings from this conference have also been published. In order to carryon this new tradition, The Third International Conference on Fibonacci Numbers and Their Applications was held in Pisa, Italy, July 25-29, 1988. This book is a result of that conference. Because of the participation at the third conference and the encouragement to hold another conference in two years, The Fourth International Conference on Fibonacci Numbers and Their Applications will take place at Winston-Salem, North Carolina, U.S.A., July 3D-August 3, 1990.
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XXIV INTRODUCTION
It is impossible to overemphasize the importance and relevance of the Fibonacci numbers to the mathematical and physical sciences as well as other areas of study. The Fibonacci numbers appear in almost every branch of mathematics, like number theory, differential equations, probability, statistics, numerical analysis, and linear algebra. They also occur in physics, biology, chemistry, and electrical engineering.
It is believed that the contents of this book will prove useful to everyone interested in this important branch of mathematics and that this material may lead to additional results on Fibonacci numbers both in mathematics and in their applications to science and engineering.
The Editors
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